Question

point p(34,42) lies on line segment ab in the cartesian plane below.
in what ratio does p divide the segment from point a?
a (6,22)
b (48,52)

Explanation:

Step1: Recall section - formula

Let the ratio in which point $P(x,y)$ divides the line - segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ be $k:1$. The section formula for the $x$ - coordinate is $x=\frac{kx_2 + x_1}{k + 1}$ and for the $y$ - coordinate is $y=\frac{ky_2 + y_1}{k + 1}$. Here, $A(6,22)$, $B(48,52)$ and $P(34,42)$. Using the $x$ - coordinate section formula: $34=\frac{k\times48+6}{k + 1}$.

Step2: Cross - multiply the $x$ - coordinate equation

Cross - multiplying the equation $34=\frac{48k + 6}{k + 1}$ gives $34(k + 1)=48k+6$. Expand the left - hand side: $34k+34 = 48k+6$.

Step3: Solve for $k$

Rearrange the equation $34k + 34=48k+6$ to get all the $k$ terms on one side. Subtract $34k$ from both sides: $34=48k - 34k+6$. Then $34 = 14k+6$. Subtract 6 from both sides: $14k=34 - 6=28$. Divide both sides by 14: $k = 2$.

Answer:

$2:1$